As was shown in [1]} for a general simplicial manifold \(X=\lbrace X_p\rbrace \) , the cohomology
of \( \lbrace \Omega ^{*,*}(X), \delta , d\rbrace \)  is isomorphic to the singular cohomology of the
geometric realization of \(X\) . On the other hand,
Dupont [2]}) introduced a related de Rham complex, \( \lbrace \Omega ^*(||X||), d\rbrace \) ,
where \(||X||\)  is the “thick” geometric realization of \(X\) ,
which has the advantage of being graded commutative. A \(q\) -form \(\omega \in \Omega ^q(||X||)\)  is a
collection \(\lbrace \omega _p\rbrace \)  of de Rham \(q\) -forms on \(\bigsqcup _p X_p\) ,
satisfying the compatibility condition
\((\varepsilon ^i \times \mathop {\rm Id}\nolimits )^*\omega _p = (\mathop {\rm Id}\nolimits \times \varepsilon _i)^*\omega _{p-1} ,\) 
